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Homework Help: Pauli Matrix question!

  1. Nov 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Pauli Spin matrices (math methods in physics question)

    Show that D can be expressed as:


    and write the [tex]d_i[/tex] in terms of D's elements, let D also be Unitary

    2. Relevant equations

    - Any 2x2 complex matrix can be written as :

    [tex]M=m_1\sigma_1+m_2\sigma_2+m_3\sigma_3+m_0I[/tex] where "I" is the identity matrix

    - Pauli spin matrix properties

    -require that D have 0 trace

    3. The attempt at a solution

    no idea where to begin honestly. please dont ding me ! This is the first day i've ever dealt with pauli spin matrices :confused:
    Last edited: Nov 29, 2012
  2. jcsd
  3. Nov 29, 2012 #2
    If it is simply given that D is a unitary matrix, then this is not true. For example, take D to be a unit matrix (which is clearly unitary). Then, as you correctly point out, it is not traceless, and cannot be represented in that way.

    What is D supposed to be?

    If D is unitary, then it may be written in an exponential form:
    D = \exp(i X)
    What property does X have if D is unitary? What if D has a unit determinant?
  4. Nov 29, 2012 #3
    thanks for the reply,

    thats exactly what I'm confused about...assume D is traceless and not unitary, does this make more sense to you? I get what you are saying, and I completely agree

    I assume from the question that D is supposed to be like M in the relevant equations section, in which the [tex]\sigma_i[/tex] correspond to the pauli matrices 1,2 and 3
  5. Nov 29, 2012 #4

    George Jones

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    Even though not every unitary 2x2 matrix is traceless, there are many unitary matrices that are tracekess, i.e., there are many unitary matrices that can be written in the form D of the original post. For example, each Pauli matrix is unitary and traceless. So is [itex]i \left( \sigma_1 + \sigma_3) \right)/\sqrt{2}[/itex]. So is ...
  6. Nov 29, 2012 #5
    Thanks for the reply, I have no idea though how to answer the original question or start it. May I have some guidance :D?
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